Yes and no is the answer. The public fountains were most people got their water from were free and available for all. However water that was delivered to private homes via the city's water mains and metered with officially calibrated nozzles was taxed, and two classes of service were available. Small houses were allowed a one half inch pipe connection to the water mains, while larger houses with baths were allowed a one and one half inch connection. The problem with that was since it was a sign of wealth to have a private supply of water people would often bribe officials to tap the aqueducts and steal water. Business would also do the same, here are some quote from Frontinus who wrote the book on the Roman Aqueduct system.

"There are extensive areas in various places where secret pipes run under the pavement all over the city. I discovered that these pipes were furnishing water by special branches to all those engaged in business in those localities through which the pipes ran, being bored for the purpose here and there by the so-called "punctures". How large an amount of water has been stolen in his manner, I estimate by means of the fact that a considerable quantity of lead has been brought in by the removal of that kind of branch pipe."

". . . a large number of landed proprietors, past whose fields the aqueducts run, tap the conduits; whence it comes that the public water courses are actually brought to a standstill by private citizens, just to water their gardens."

So in conclusion water was free unless you wanted it in your house in which case it was expensively taxed and monitored to the extent that people would rather bribe officials then actually pay the fees.

http://www.uvm.edu/~rrodgers/Frontinus.html

*edit Thanks for the Gold stranger.